There are a few things going on here and if you can provide a bit more detail I may be able to offer a better answer.
What kind of headphones are you using?
What kind of D/A are you using (the headphone jack on your phone/computer or an interface)?
Do you experience this with speakers as well?
For reference, what are the tracks in question?
To address the Spotify issue specifically I have noticed that some of there tracks dont sound great but its all way better than youtube or pandora so ill take it considering the amount of stuff you can stream. I have noticed some high end bias when using stock apple headphones but I account this much more to their design than the streaming service. I spend most of my day with a pair of AKG 240's on my head and have never had an issue with Spotify (which I also use all day) through those cans. All that being said they are streaming a compressed (mp3 or something) and compressed formats can leave things to be desired and cause not so great tone.
You are in a way correct that there could be some frequencies outside your hearing range but in general AFAIK most consumer playback devices cant really reproduce these frequencies very well as they are optimized (and sometimes filtered) to only produce frequencies in the human spectrum. Especially single cone speakers like the ones found in headphones, the issue being that speakers that produce good low end freq are usually not good at producing high freq so engineers chose something somewhat in the middle and optimize it across the human hearing range.
On the other hand high freq sound can and does cause discomfort to humans. Depending on how much music you listen discomfort can also be caused by simply listening to music for a long continuous period of time. Its common in the workforce these days to be allowed to listen to music during the work day. 8 more or less continuous hours of noise can really strain the ears. This may be a bit of a drastic take but its worth checking out. This article is more related to mixing but touches on some similar points. Here is an article that directly address your points.